Harmonic Oscillator with fermions

In summary, a Harmonic Oscillator with fermions is a quantum mechanical system that describes the behavior of fermions in a harmonic potential. It differs from a regular Harmonic Oscillator in that it describes the behavior of particles with half-integer spin, rather than particles with integer spin. Studying this system helps us understand the behavior of fermions in a harmonic potential, which has important implications in various fields of physics. The energy spectrum of a Harmonic Oscillator with fermions is different from that of a regular Harmonic Oscillator due to the Pauli exclusion principle, and in most cases, the system cannot be solved exactly.
  • #1
milesAhead
2
0
Hello...
We have 3 fermions (s=1/2) at the ground state of a harmonic oscillator moving over the x-axis with a the classic hamiltonian for a three particle oscillator :
H =(1/2m)*(P1)^2 +((1/2)*m(w^2)((x1)^2)) +(1/2m)*(P2)^2 +((1/2)*m(w^2)((x2)^2)) +(1/2m)*(P3)^2 +((1/2)*m(w^2)((x3)^2))

we have a pertubation between fermions that is:
W=g*m*((w^2)/h)*(S1z*x2*x3+S2z*x3*x1+S3z*x1*x2).

And it is asked the chande of the ground state's energy due to the pertubation.
 
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  • #2
With S1z,S2z,S3z the spin (z) of each fermion
 
  • #3


I would first clarify that the Hamiltonian provided is for a classical harmonic oscillator system, not a quantum one. To consider the change in energy of the system due to the perturbation, we need to use the quantum mechanical formalism and consider the wavefunction of the system.

Assuming that the three fermions are identical and indistinguishable, we can write the wavefunction as a product of single-particle wavefunctions, each representing the state of one fermion in the oscillator:

Ψ(x1,x2,x3) = φ1(x1)φ2(x2)φ3(x3)

where φi(xi) is the wavefunction of the ith fermion in the oscillator.

To calculate the change in energy, we can use the perturbation theory and consider the first-order correction to the ground state energy:

ΔE = ⟨Ψ|W|Ψ⟩

where W is the perturbation operator and Ψ is the ground state wavefunction.

Substituting the perturbation operator and the wavefunction, we get:

ΔE = g*m*((w^2)/h)*⟨φ1φ2φ3|S1z*x2*x3+S2z*x3*x1+S3z*x1*x2|φ1φ2φ3⟩

Since the three fermions are identical, the expectation value of any operator involving only one fermion will be the same for all three fermions. Therefore, we can write:

ΔE = 3*g*m*((w^2)/h)*⟨φ1|S1z|x1|φ1⟩*⟨φ2|x2|φ2⟩*⟨φ3|x3|φ3⟩

Using the commutation relation [S1z,x1] = iħ, we can simplify the above expression to:

ΔE = 3*g*m*((w^2)/h)*ħ*ħ*ħ

Therefore, the change in energy of the ground state due to the perturbation is directly proportional to the coupling constant g and the square of the oscillator frequency w. This result is consistent with the classical result, where the perturbation term also depends on g and w^2.

In conclusion, the perturbation between fermions in a harmonic oscillator system results in a change in the ground state energy, which can be calculated using the quantum mechanical formalism and perturbation theory.
 

1. What is a Harmonic Oscillator with fermions?

A Harmonic Oscillator with fermions is a quantum mechanical system that describes the behavior of fermions (particles with half-integer spin) in a harmonic potential. It is a common model used in condensed matter physics and quantum field theory.

2. How does a Harmonic Oscillator with fermions differ from a regular Harmonic Oscillator?

While both systems follow the same mathematical model, the main difference is in the types of particles that are being described. A regular Harmonic Oscillator typically describes the behavior of bosons (particles with integer spin), while a Harmonic Oscillator with fermions describes the behavior of fermions.

3. What is the significance of studying the Harmonic Oscillator with fermions?

Studying the Harmonic Oscillator with fermions helps us understand how particles with half-integer spin behave in a harmonic potential. This has important implications in various fields of physics, such as condensed matter physics, quantum computing, and particle physics.

4. How is the energy spectrum of a Harmonic Oscillator with fermions different from that of a regular Harmonic Oscillator?

The energy spectrum of a Harmonic Oscillator with fermions is different from that of a regular Harmonic Oscillator due to the Pauli exclusion principle. This principle states that no two identical fermions can occupy the same quantum state, leading to a different energy level structure for fermions compared to bosons.

5. Can the Harmonic Oscillator with fermions be solved exactly?

In most cases, the Harmonic Oscillator with fermions cannot be solved exactly. However, there are certain special cases, such as the one-dimensional case, where the system can be solved exactly. In other cases, numerical methods or approximations are used to solve the system.

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